Journal of Fuzzy Set Valued Analysis 2014 (2014) 1-8
Available online at www.ispacs.com/jfsva Volume 2014, Year 2014 Article ID jfsva-00190, 8 Pages doi:10.5899/2014/jfsva-00190 Research Article
On the global solution of a fuzzy linear system Tofigh Allahviranloo1∗ , Arjan Skuka2 , Sahar Tahvili3 (1) Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran. (2) Department of Software Engineering, Faculty of Engineering, Izmir University, Izmir, Turkey. (3) Division of Applied Mathematics, The School of Education, Culture and Communication, Malardalen ¨ university, Vaster ¨ as-Sweden. ˙ c Tofigh Allahviranloo, Arjan Skuka and Sahar Tahvili. This is an open access article distributed under the Creative Commons Copyright 2014 ⃝ Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract The global solution of a fuzzy linear system contains the crisp vector solution of a real linear system. So discussion about the global solution of a n × n fuzzy linear system Ax˜ = b˜ with a fuzzy number vector b in the right hand side and crisp a coefficient matrix A is considered. The advantage of the paper is developing a new algorithm to find the solution of such system by considering a global solution based upon the concept of a convex fuzzy numbers. At first the existence and uniqueness of the solution are introduced and then the related theorems and properties about the solution are proved in details. Finally the method is illustrated by solving some numerical examples. Keywords: Fuzzy linear system, Algebraic solution, Fuzzy number.
1 Introduction Fuzzy linear systems have many applications in science, such as control problems, information, physics, statistics, engineering, economics, finance and even social science. In the 1990s, Buckley et al. [11, 12, 13] investigated the mentioned systems in analytical form. Subsequently, Friedman et al. [14] considered a fuzzy linear system as follows, a11 x˜1 + a12 x˜2 + . . . + a1n x˜n = b˜ 1 a21 x˜1 + a22 x˜2 + . . . + a2n x˜n = b˜ n (1.1) .. .. .. . . . an1 x˜1 + an2 x˜2 + . . . + ann x˜n = b˜ n where the coefficient matrix A = (ai j ) is a crisp matrix and b˜ = (bi ) is a fuzzy vector for 1 6 i, j 6 n. They used the embedding method and replaced the original fuzzy linear system by a crisp linear system with a nonnegative coefficient matrix S, which is singular even if A is nonsingular. They also presented conditions for the existence of a unique fuzzy solution to the system. In [3, 4], Allahviranloo has investigated the various numerical methods (Jacobi, Gauss Seidel) for solving such fuzzy linear systems for the first time. Also he proposed the Adomian and Homotopy methods for solving fuzzy linear systems in [7, 9]. The several different numerical techniques for solving them are proposed in [1, 5, 6]. Also in [2] Allahviranloo et. al. have shown that the proposed method by [14] has no weak solution generally. Recently, he and Salahshour in [10] proposed a simple and practical method to obtain fuzzy ∗ Correspondence
author Email address:
[email protected], Tel: +989123508816.
Journal of Fuzzy Set Valued Analysis http://www.ispacs.com/journals/jfsva/2014/jfsva-00190/
Page 2 of 8
symmetric solutions of fuzzy linear systems. The algebraic solution of such systems and its properties are discussed in [8]. Also Ghanbari and his colleague in [15] have proposed an approach for computing the general compromised solution of an L-R fuzzy linear system by use of a ranking function when the coefficient matrix is a crisp m × n matrix. Zengfeng et. al in [19], worked on the perturbation analysis of fuzzy linear systems. Wang et. al in [18], considered Jacobi and Gauss Seidel iteration methods for solving the fuzzy linear system. Summarised the structure of the paper is as follows: In section 2, some basic definitions and notions about fuzzy concepts are brought. In section 3, we are going to represent the algebraic solution and its properties. In section 4 the method and main results are discussed. In section 5, based on the proposed method, an algorithm for solving some numerical examples is designed. Finally, conclusions are drawn in section 6. 2 Preliminaries In this section some basic definitions and notations are brought. ( ) Definition 2.1. A fuzzy number x˜ is shown as an ordered pair of functions x˜ = x(r), x(r) ¯ where: i) x(r) is a left-continuous bounded monotonic increasing function. ii) x(r) ¯ is a left-continuous bounded monotonic decreasing function. iii) x(r) ≤ x(r), ¯ r ∈ [0, 1]. The set of fuzzy numbers is shown in the form of E 1 . Definition 2.2. If x˜ ∈ E 1 then the support is defined as follows: ¯ supp x˜ = {x ∈ R | µx˜ (x) > 0} = [x(0), x(0)]. And if x˜ ∈ E n then:
n
supp x˜ = ∏ [x j (0), x j (0)]. j=1
Definition 2.3. A fuzzy number in x˜ ∈ E 1 is called convex if all r−level sets are convex for each r. Also a fuzzy number vector x˜ ∈ E n is called convex when ∏nj=1 [x j (r), x j (r)] is a convex polygon. Definition 2.4. A fuzzy number vector is shown as x˜ = (x˜1 , x˜2 , . . . , x˜n )t ∈ E n in which x˜i is a fuzzy number in E 1 . It could be written as: ( ) (( )t ( )t ) x˜ = x(r), x(r) = x1 (r), . . . , xn (r) , x1 (r), . . . , xn (r) (2.2) where for 0 ≤ r ≤ 1, ( )t x1 (r), . . . , xn (r) = min{ u = (u1 , u2 , . . . , un )t | u ∈ ∏nj=1 [x j (r), x j (r)], b ∈ ∏ni=1 [bi (r), bi (r)], Au = b}, (
)t x1 (r), . . . , xn (r) = max{ u = (u1 , u2 , . . . , un )t | u ∈ ∏nj=1 [x j (r), x j (r)], b ∈ ∏ni=1 [bi (r), bi (r)], Au = b}.
Also based on the extension principle, the membership function of an arbitrary vector x = (x1 , . . . , xn )t ∈ Rn is defined as the following: µx˜ (x) = min {µx˜ j (x j )} 1≤ j≤n
in which µx˜ j (x), j = 1, 2, . . . , n is the membership function of fuzzy number x˜ j .
International Scientific Publications and Consulting Services
Journal of Fuzzy Set Valued Analysis http://www.ispacs.com/journals/jfsva/2014/jfsva-00190/
Page 3 of 8
3 Fuzzy linear system Consider the n × n fuzzy linear system (1.1). It can be written in matrix form as follows, Ax˜ = b˜
(3.3)
We denote the set of all crisp solutions of system (3.3) by χ∃ . So then, χ∃ = {x ∈ Rn | ∃b ∈ b˜ ; Ax = b}. Definition 3.1. The vector x˜ ∈ E n will be called an algebraic solution of system (3.3) if: n
∑ ai j x˜ j = b˜ i ⇒
j=1
n
∑ ai j [x j (r), x j (r)] = [b j (r), b j (r)],
i = 1, 2, . . . , n.
(3.4)
j=1
Note 1: If the system (3.3) has the algebraic solution x, ˜ then x˜ ⊆ χ∃ . Note 2: If we just deal with the algebraic solution, we should consider the following 4 cases: 1. From (3.4) it is clear that the algebraic solution is obtained by the exact equality between two fuzzy numbers. a, ˜ b˜ ∈ E 1 , a˜ = b˜ i.e a(r) = b(r), a(r) = b(r) Using the definition 3.1, each equation is transformed to the following equations: n
∑ ai j x j (r)
= b(r),
i = 1, 2, . . . n
∑ ai j x j (r)
=
i = 1, 2, . . . n
j=1 n
b(r),
j=1
So, two n × n crisp systems are produced. 2. In the proposed methods to find the algebraic solution, it is necessary that 2n × 2n crisp system or two n × n crisp systems are solved. 3. To find the algebraic solution, the interval arithmetic is employed. Since the calculate operations on intervals are based on the extension principle, it causes the extension of the width of intervals. Therefore, in application there is usually no algebraic solution. See [19, 21] for more information. 4. Since the algebraic solution is a subset of χ∃ , it may does not include all the crisp vectors. For description see the following example: Example 3.1. [14], Consider the 2 × 2 fuzzy linear system as following: { x1 − x2 = (r, 2 − r) x1 + 3x2 = (4 + r, 7 − 2r)
(3.5)
The solution of the system is as: x1 (r) = 1.375 + 0.625r
, x1 (r) = 2.875 − 0.875r
x2 (r) = 0.875 + 0.125r It is seen that, if x˜ = (x˜1 , x˜2 )t and b˜ = (b˜ 1 , b˜ 2 )t then:
, x2 (r) = 1.375 − 0.375r
supp x˜ = [1.375, 2.875] × [0.875, 1.375] supp b˜ = [0, 2] × [4, 7] Now, if we choose vector b˜ = (0.1, 4.1) ∈ supp b˜ then we have: { xˆ1 − xˆ2 = 0.1 ⇒ xˆ = (1.1, 1)t xˆ1 + 3xˆ2 = 4.1 It is clear that, xˆ ̸∈ supp x˜ and thus, µx˜ (x) ˆ = 0.
International Scientific Publications and Consulting Services
Journal of Fuzzy Set Valued Analysis http://www.ispacs.com/journals/jfsva/2014/jfsva-00190/
Page 4 of 8
4 The proposed method In this section we are going to introduce the unique algebraic solution of system (3.3). Let A be nonsingular. )t ( Definition 4.1. The vector X˜ = x˜1 , x˜2 , . . . , x˜n in which x˜i = (xi (r), xi (r)), i = 1, . . . , n is called a global solution of system (3.3) whenever for r ∈ [0, 1]: } { ( )t xi (r) = inf xi (r) | x(r) = x1 (r), . . . , xn (r) ∈ χ∃ { ( )t } xi (r) = sup xi (r) | x(r) = x1 (r), . . . , xn (r) ∈ χ∃ Theorem 4.1. If in system (3.3), matrix A is nonsingular, then there is a unique global fuzzy number vector solution. Proof. The proof of uniqueness is clear and for existence it is structural. We define two sets of vectors as the following for arbitrary and fixed 0 ≤ r ≤ 1: { } ( )t I(r) = v(r) ∈ Rn | v(r) = v1 (r), . . . , vn (r) , v j (r) ∈ {b j (r), b j (r)} { } ( )t χ (r) = x(r) ∈ Rn | x(r) = x1 (r), . . . , xn (r) , Ax(r) = v(r) ∈ I(r) The set I(r) has 2n elements. Thus the set χ (r) is obtained by solving 2n crisp systems. It is clear that, in definition 4.1, ”inf” and ”sup” is replaced by ”min” and ”max”, as the following. So for j = 1, . . . , n, { } (k) (k) (k) x j (r) = min n x j (r) | x(k) = (x1 , . . . , xn )t , x(k) ∈ χ (r) (4.6) 1≤k≤2
{ } (k) (k) (k) x j (r) = max n x j (r) | x(k) = (x1 , . . . , xn )t , x(k) ∈ χ (r) 1≤k≤2
(4.7)
in which, 0 ≤ r ≤ 1 is arbitrary and fixed. ( ) Now, we show that x˜ j = x j (r), x j (r) for r ∈ [0, 1], is a fuzzy number. Considering structural proof, it is clear that x j (r) ≤ x j (r). Assume that A−1 = [ti j ]ni, j=1 . For any r, the vector v(k) (r) ∈ I(r) corresponds to the vector x(k) (r) ∈ χ (r), and we have: Ax(k) (r) = (k)
⇒ xi (r) =
v(k) (r) ⇒ x(k) (r) = A−1 v(k) (r) n
∑ ti j v j
(k)
(r)
j=1 n
=
∑ ti j v j
(k)
n
(r) +
ti j ≥0
∑ ti j v j
(k)
(r)
(4.8)
ti j
